# Definition:Dual Ordering/Dual Ordered Set

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## Contents

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $\succeq$ be the dual ordering of $\preceq$.

The ordered set $\left({S, \succeq}\right)$ is called the **dual ordered set** (or just **dual**) **of $\left({S, \preceq}\right)$**.

That it indeed is an ordered set is a consequence of Dual Ordering is Ordering.

## Also known as

A quite popular alternative for **dual ordered set** is **opposite poset**.

However, since this use conflicts with ProofWiki's definition of a partially ordered set, **dual ordered set** is the name to be used.

**Inverse ordered set** can also be encountered.

## Also see

## Sources

- 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S \text I.2$